That's only possible with custom modification and you'll find some examples below.

WORDPRESS ARCHIVES (Not a solution)
Normally WordPress creates archive pages and it would be: http://themes.devatic.com/daisho/portfolio/identity
But:
1) Archives are disabled for portfolio
2) And even if they were enabled it would look like this: http://themes.devatic.com/daisho/tag/letterpress/ (archive.php) and not like a thumbnails grid (template-portfolio.php).
That's dead end then because you would need archive-portfolio.php file with almost the same content like in template-portfolio.php.

This will not generate an archive page but indeed, it allows to select different category on start and you can even manipulate buttons or create custom ones with conditional behaviors. This doesn't cover URL change.

ISOTOPE HASH CHANGE (Solution #2)
Lastly, Isotope (portfolio script) has an excellent API and it gives you another way to do what you're looking for. Technically it will not have anything in common wih Wordpress' archives and links but visually it will look and work like expected for your visitors. Isotope's documentation on URL change (http://isotope.metafizzy.co/docs/hash-history-jquery-bbq.html), using Ben Alman's jQuery BBQ.

Let me know if you need more information about that. Help of third party developers may be useful here as it requires custom code.

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UPDATE with the full code:

Place the following script at the very bottom of /wp-content/themes/daisho/js/portfolio.js: